\section{1.6} 
\begin{frame}[allowframebreaks]{1.6. }

\vspace{-0.4cm}

1.6. $\mathcal{D}$-modules and connections. 

Let $M$ be a quasi-coherent $\mathcal{O}_X$-module. 

A connection on $M$ is an operation $\xi \mapsto \nabla_\xi$ of $\Theta_X$ on $M$ which satisfies
\begin{equation}
\nabla_{a\xi + b\eta} = a\nabla_\xi + b\nabla_\eta \quad (a,b \in \mathcal{O}_X, \xi,\eta \in \Theta_X)
\end{equation}
\begin{equation}
\nabla_\xi(f.m) = (\xi f).m + f\nabla_\xi m \quad (\xi \in \Theta_X, f \in \mathcal{O}_X), m \in M);
\end{equation}
the latter can also be written as
\begin{equation}
[\xi, f]m = (\xi f).m
\end{equation}

As usual, the curvature of a connection is given by $[\nabla_\xi, \nabla_\eta] - \nabla_{[\xi,\eta]}$ $(\xi,\eta \in \Theta_X)$. 

Therefore the connection extends to a $\mathcal{D}_X$-module structure if and only if the curvature is zero i.e. 
if
\begin{equation}
[\nabla_\xi, \nabla_\eta] = \nabla_{[\xi,\eta]} \quad (\xi,\eta \in \Theta_X).
\end{equation}

In other words, a $\mathcal{D}_X$-module can also be defined as a $\mathcal{O}_X$-module endowed with a flat connection.

We are accustomed to that notion in case $M$ is locally free of finite rank, i.e. on case where $M$ is the sheaf of germs of regular sections of an algebraic vector bundle. 

In fact, if suffices for this that $M$ be $\mathcal{O}_X$-coherent:

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